23. Drawing Cards Two cards are to be randomly selected without replacement from a shuffled deck. Find the probability of getting an ace on the first card and a spade on the second card.

1. Probability of at Least One You want to find the probability of getting at least 1 defect when 10 heart pacemakers are randomly selected and tested. What do you know about the exact number of defects if “at least one” of the 10 items is defective?
k k k k PsB k Ad PsA k Bd k 2 4-5 Multiplication Rule: Complements and Conditional Probability 171 the 300 subjects used marijuana. We get

b. Here we want P(marijuana use positive). If we assume that the person selected tested positive, we are dealing with the 143 subjects in the first row of Table 4-1. Among those 143 subjects, 119 used marijuana, so Again, the same result can be found by applying the formula for conditional probability:
By comparing the results from parts

(a) and (b), we see that P(positive marijuana use) is not the same as P(marijuana use positive).
INTERPRETATION The first result of P(positive marijuana use) 5 0.975 indicates that a marijuana user has a 0.975 probability of testing positive. The second result of P(marijuana use positive) 5 0.832 indicates that for someone who tests positive, there is an 0.832 probability that this person actually used marijuana.
k k k k 5 119>300 143>300 5 0.832 Psmarijuana use k positived 5 Pspositive and marijuana used Pspositived Psmarijuana use k positived 5 119 143 5 0.832 k 5 119>300 122>300 5 0.975 Pspositive k marijuana used 5 Psmarijuana use and positived Psmarijuana used Composite Sampling The U.S. Army once tested for syphilis by giving each inductee an individual blood test that was analyzed separately.
One researcher suggested mixing pairs of blood samples. After the mixed pairs were tested, syphilitic inductees could be identified by retesting the few blood samples that were in the pairs that tested positive. The total number of analyses was reduced by pairing blood specimens, so why not put them in groups of three or four or more? Probability theory was used to find the most efficient group size, and a general theory was developed for detecting the defects in any population. This technique is known as composite sampling.